332 ANALYTICAL MHCIIANICS 8. In the preceding problem suppose both of the particles to be i and show that they make n y ///I~~- vibrations per second. 9. A string which connects two particles of equal mass passes throi a small hole in a smooth horizontal table. One of the particles ha: vertically while the other, which is on the table at a distance I) from hole, is given a velocity Vgl) in a direction perpendicular to the stri Show that the suspended particle, will be in equilibrium and that if i slightly disturbed it will vibrate with a period of 2 TT \-~c~- 10. The piston of a cylinder, which is in a vertical position, is in ec librium under the action of it-s weight and the upward pressure of the in the cylinder. Show that when the cylinder is given a small dinpla ment it will vibrate with a period equal to 2 w y ', where // is the heij of the piston above the base* of the cylinder wheu the former is at ita eq librium position. Assume Boyle's law to hold. 11. In illustrative problem 2 (p. 32S) take the mass of the* string ii account and obtain the expression for the period of vibration. 12. In problem G take the mass of the spring into account and obi an expression for the period, 13. In problem 7 take the mass of the spring into account and find expression for the period of vibrations. 14. In problem 8 take the mass of the spring into account and find i expression for the period. 15. A particle is placed at, the center of a smooth horizontal table; t particles of the same mass as the first one art* suspended by means strings of negligible mass, each of which passes over a .smooth pulley the middle point of one of the edges of the table and is attached to 1 first particle. The particle at the center is given a small displaeem< at right angles to the stringn. Bhow that it jwforms small oseillati< with a period of 2r y ~, where a is the distance between the two pulle, 16. A particle rests at the center of a square table winch is smooth a horizontal. Four particles are suspended by means of strings each which passes over an edge of the table and is connected to the particle. the table. Find the period with which the system will vibrate when 1 particle which is on the table Ls displaced along one of the .stringy. T particles have equal mass. Neglect the mass of the strings.